/* In this case we let H denote a copy of PSL(2,19), L denote the subgroup 19.9, and let G denote E6(2^18).
   We give matrices defining L, an element of order 19 and an element of order 9. We construct L to lie in
   normalizer of a torus NGT, as constructed using the GroupOfLieType command. We give H as L and a matrix
   h1, chosen at random. A posteriori, one could produce H as a subgroup of E6 and show that no other
   overgroup of L preserves the E6 trilinear form, but we prefer a less synthetic proof.

   We also prove that the copies of H lie in J3.
*/

q:=2^18;
F<zz>:=GF(q);
z:=zz^13797;

l1:=GL(27,F)![[0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],
[0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0],
[0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],
[0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],
[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0]];

l2:=GL(27,F)![[zz^55188,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,zz^151767,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,zz^206955,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,zz^27594,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,zz^220752,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,zz^234549,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,zz^179361,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,zz^165564,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,zz^110376,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,zz^248346,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,zz^193158,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,zz^41391,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,zz^137970,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,zz^248346,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^137970,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^193158,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^82782,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^13797,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^27594,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^206955,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^110376,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^41391,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^165564,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^96579,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^179361,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^124173,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^68985]];

L:=sub<GL(27,F)|l1,l2>;

h1:=GL(27,F)![[zz^156475,zz^258988,0,0,zz^185200,zz^75328,zz^43914,zz^112914,0,0,zz^262095,zz^140253,0,zz^16272,zz^194673,0,zz^177139,zz^225661,zz^140505,zz^254334,zz^164007,0,0,zz^101614,0,zz^202717,zz^1177],
[zz^65830,zz^61027,0,0,zz^177139,zz^128881,zz^195696,zz^110328,0,0,zz^249861,zz^140493,0,zz^85317,zz^181836,0,zz^115411,zz^227095,zz^116364,zz^191601,zz^98862,0,0,zz^249523,0,zz^156676,zz^244108],
[0,0,zz^2700,zz^112185,0,0,0,0,zz^127791,zz^153117,0,0,zz^90585,0,0,zz^235224,0,0,0,0,0,zz^19197,zz^10800,0,zz^48627,0,0],
[0,0,zz^208764,zz^166914,0,0,0,0,zz^195858,zz^145773,0,0,zz^138645,0,0,zz^121176,0,0,0,0,0,zz^127791,zz^186597,0,zz^143370,0,0],
[zz^116215,zz^39169,0,0,zz^101614,zz^24439,zz^179313,zz^84297,0,0,zz^223227,zz^16335,0,zz^47379,zz^237084,0,zz^249523,zz^4708,zz^39834,zz^112659,zz^85305,0,0,zz^216514,0,zz^144313,zz^184270],
[zz^240892,zz^225460,0,0,zz^258988,zz^211864,zz^181644,zz^99102,0,0,zz^140508,zz^81225,0,zz^167079,zz^82734,0,zz^61027,zz^147529,zz^209430,zz^154302,zz^102567,0,0,zz^39169,0,zz^97756,zz^253381],
[zz^176806,zz^259603,0,0,zz^119047,zz^148972,zz^9559,zz^142879,0,0,zz^33937,zz^125350,0,zz^38236,zz^32302,0,zz^106813,zz^103741,zz^157606,zz^25372,zz^198193,0,0,zz^80179,0,zz^97489,zz^163021],
[zz^11257,zz^201829,0,0,zz^51625,zz^94024,zz^170473,zz^89125,0,0,zz^32302,zz^50950,0,zz^47230,zz^148612,0,zz^38788,zz^245551,zz^94357,zz^170599,zz^28771,0,0,zz^94036,0,zz^241711,zz^83692],
[0,0,zz^58806,zz^30294,0,0,0,0,zz^143370,zz^222561,0,0,zz^222102,0,0,zz^49194,0,0,0,0,0,zz^48627,zz^249021,0,zz^259146,0,0],
[0,0,zz^70335,zz^228555,0,0,0,0,zz^208764,zz^675,0,0,zz^93582,0,0,zz^103815,0,0,0,0,0,zz^2700,zz^88182,0,zz^58806,0,0],
[zz^105250,zz^24031,0,0,zz^135367,zz^80242,zz^6343,zz^239257,0,0,zz^38236,zz^101488,0,zz^135748,zz^47230,0,zz^204412,zz^80227,zz^129208,zz^152944,zz^106138,0,0,zz^166474,0,zz^11002,zz^117538],
[zz^24799,zz^218197,0,0,zz^232009,zz^62350,zz^139147,zz^37153,0,0,zz^142879,zz^87817,0,zz^239257,zz^89125,0,zz^229423,zz^14707,zz^203800,zz^143809,zz^108154,0,0,zz^203392,0,zz^232006,zz^204352],
[0,0,zz^228555,zz^180036,0,0,0,0,zz^166914,zz^52191,0,0,zz^172800,0,0,zz^101979,0,0,0,0,0,zz^112185,zz^100197,0,zz^30294,0,0],
[zz^11194,zz^11254,0,0,zz^111286,zz^258580,zz^162409,zz^143809,0,0,zz^25372,zz^87490,0,zz^152944,zz^170599,0,zz^48553,zz^160438,zz^188920,zz^18706,zz^254689,0,0,zz^231754,0,zz^94039,zz^245791],
[zz^148204,zz^66382,0,0,zz^259600,zz^132844,zz^115084,zz^203800,0,0,zz^157606,zz^158110,0,zz^129208,zz^94357,0,zz^235459,zz^25054,zz^70162,zz^188920,zz^115285,0,0,zz^158929,0,zz^66430,zz^107581],
[0,0,zz^675,zz^52191,0,0,0,0,zz^145773,zz^214191,0,0,zz^253746,0,0,zz^196776,0,0,0,0,0,zz^153117,zz^154467,0,zz^222561,0,0],
[zz^121951,zz^253381,0,0,zz^1177,zz^102418,zz^180876,zz^85257,0,0,zz^43926,zz^27546,0,zz^260586,zz^226740,0,zz^244108,zz^190003,zz^250629,zz^126696,zz^250884,0,0,zz^184270,0,zz^211663,zz^199501],
[zz^46300,zz^240892,0,0,zz^156475,zz^64747,zz^53631,zz^167847,0,0,zz^57711,zz^212943,0,zz^248298,zz^154305,0,zz^65830,zz^235726,zz^29109,zz^154242,zz^195693,0,0,zz^116215,0,zz^18832,zz^121951],
[zz^52645,zz^221662,0,0,zz^20959,zz^218149,zz^198997,zz^108154,0,0,zz^198193,zz^231394,0,zz^106138,zz^28771,0,zz^217957,zz^120610,zz^115285,zz^254689,zz^18505,0,0,zz^204400,0,zz^63859,zz^107836],
[zz^69895,zz^200320,0,0,zz^259348,zz^66442,zz^232327,zz^87817,0,0,zz^125350,zz^74824,0,zz^101488,zz^50950,0,zz^259588,zz^149164,zz^158110,zz^87490,zz^231394,0,0,zz^135430,0,zz^93016,zz^146641],
[zz^172726,zz^38596,0,0,zz^163009,zz^207865,zz^74020,zz^139147,0,0,zz^9559,zz^232327,0,zz^6343,zz^170473,0,zz^52648,zz^121630,zz^115084,zz^162409,zz^198997,0,0,zz^36265,0,zz^245743,zz^37828],
[0,0,zz^88182,zz^100197,0,0,0,0,zz^186597,zz^154467,0,0,zz^43200,0,0,zz^194508,0,0,0,0,0,zz^10800,zz^76788,0,zz^249021,0,0],
[0,0,zz^93582,zz^172800,0,0,0,0,zz^138645,zz^253746,0,0,zz^45009,0,0,zz^209655,0,0,0,0,0,zz^90585,zz^43200,0,zz^222102,0,0],
[zz^18832,zz^97756,0,0,zz^202717,zz^52966,zz^126648,zz^112911,0,0,zz^240537,zz^236064,0,zz^154050,zz^122616,0,zz^156676,zz^212794,zz^209478,zz^237087,zz^206907,0,0,zz^144313,0,zz^79627,zz^211663],
[0,0,zz^103815,zz^101979,0,0,0,0,zz^121176,zz^196776,0,0,zz^209655,0,0,zz^250155,0,0,0,0,0,zz^235224,zz^194508,0,zz^49194,0,0],
[zz^64747,zz^211864,0,0,zz^75328,zz^56365,zz^88770,zz^205398,0,0,zz^29877,zz^209490,0,zz^223290,zz^237072,0,zz^128881,zz^60223,zz^13749,zz^139485,zz^99054,0,0,zz^24439,0,zz^52966,zz^102418],
[zz^235726,zz^147529,0,0,zz^225661,zz^60223,zz^2535,zz^157755,0,0,zz^246789,zz^30069,0,zz^223275,zz^126456,0,zz^227095,zz^11575,zz^168102,zz^41343,zz^1515,0,0,zz^4708,0,zz^212794,zz^190003]];

h2:=GL(27,F)![[zz^230570,zz^247235,0,0,zz^37010,zz^212276,zz^139389,zz^9099,0,0,zz^244128,zz^16509,0,zz^225189,zz^51000,0,zz^200645,zz^57542,zz^76548,zz^9483,zz^202761,0,0,zz^135851,0,zz^134243,zz^216308],
[zz^54077,zz^174980,0,0,zz^200645,zz^202976,zz^91881,zz^92361,0,0,zz^106188,zz^235968,0,zz^21360,zz^128610,0,zz^149648,zz^101900,zz^133656,zz^230355,zz^237261,0,0,zz^202511,0,zz^250700,zz^175634],
[0,0,zz^2700,zz^112185,0,0,0,0,zz^127791,zz^153117,0,0,zz^90585,0,0,zz^235224,0,0,0,0,0,zz^19197,zz^10800,0,zz^48627,0,0],
[0,0,zz^208764,zz^166914,0,0,0,0,zz^195858,zz^145773,0,0,zz^138645,0,0,zz^121176,0,0,0,0,0,zz^127791,zz^186597,0,zz^143370,0,0],
[zz^230168,zz^62675,0,0,zz^135851,zz^12686,zz^161346,zz^202767,0,0,zz^170001,zz^174663,0,zz^64671,zz^173127,0,zz^202511,zz^78803,zz^78588,zz^251058,zz^180780,0,0,zz^148040,0,zz^19118,zz^16151],
[zz^115697,zz^37412,0,0,zz^247235,zz^43745,zz^57900,zz^194577,0,0,zz^36693,zz^119979,0,zz^23406,zz^64767,0,zz^174980,zz^79055,zz^156204,zz^90345,zz^119859,0,0,zz^62675,0,zz^50744,zz^25475],
[zz^244340,zz^127847,0,0,zz^73139,zz^259430,zz^203228,zz^236903,0,0,zz^170885,zz^78338,0,zz^26483,zz^106397,0,zz^197342,zz^114554,zz^9416,zz^48878,zz^232430,0,0,zz^261155,0,zz^86840,zz^71123],
[zz^141644,zz^155921,0,0,zz^142154,zz^161558,zz^2354,zz^226073,0,0,zz^106397,zz^244619,0,zz^161183,zz^136859,0,zz^219764,zz^93866,zz^117863,zz^204836,zz^243902,0,0,zz^2138,0,zz^252524,zz^73043],
[0,0,zz^58806,zz^30294,0,0,0,0,zz^143370,zz^222561,0,0,zz^222102,0,0,zz^49194,0,0,0,0,0,zz^48627,zz^249021,0,zz^259146,0,0],
[0,0,zz^70335,zz^228555,0,0,0,0,zz^208764,zz^675,0,0,zz^93582,0,0,zz^103815,0,0,0,0,0,zz^2700,zz^88182,0,zz^58806,0,0],
[zz^59342,zz^114560,0,0,zz^54200,zz^210629,zz^143291,zz^51209,0,0,zz^26483,zz^195512,0,zz^159254,zz^161183,0,zz^112514,zz^147761,zz^163445,zz^105932,zz^37664,0,0,zz^155825,0,zz^121460,zz^128351],
[zz^135257,zz^23588,0,0,zz^100253,zz^73163,zz^92135,zz^230822,0,0,zz^236903,zz^122054,0,zz^51209,zz^226073,0,zz^183515,zz^4058,zz^192047,zz^257762,zz^131660,0,0,zz^31778,0,zz^140108,zz^123185],
[0,0,zz^228555,zz^180036,0,0,0,0,zz^166914,zz^52191,0,0,zz^172800,0,0,zz^101979,0,0,0,0,0,zz^112185,zz^100197,0,zz^30294,0,0],
[zz^192170,zz^181499,0,0,zz^100637,zz^86966,zz^150656,zz^257762,0,0,zz^48878,zz^161585,0,zz^105932,zz^204836,0,zz^59366,zz^114530,zz^120446,zz^112730,zz^129494,0,0,zz^80069,0,zz^224426,zz^51182],
[zz^238733,zz^247358,0,0,zz^167702,zz^86936,zz^189179,zz^192047,0,0,zz^9416,zz^32915,0,zz^163445,zz^117863,0,zz^224810,zz^155441,zz^23150,zz^120446,zz^209309,0,0,zz^169742,0,zz^133964,zz^218039],
[0,0,zz^675,zz^52191,0,0,0,0,zz^145773,zz^214191,0,0,zz^253746,0,0,zz^196776,0,0,0,0,0,zz^153117,zz^154467,0,zz^222561,0,0],
[zz^145457,zz^25475,0,0,zz^216308,zz^216371,zz^37203,zz^32031,0,0,zz^242112,zz^9579,0,zz^37197,zz^244032,0,zz^175634,zz^178250,zz^126885,zz^222171,zz^147069,0,0,zz^16151,0,zz^23615,zz^74306],
[zz^140324,zz^115697,0,0,zz^230570,zz^258416,zz^92385,zz^44103,0,0,zz^153186,zz^230235,0,zz^230331,zz^50490,0,zz^54077,zz^188714,zz^147579,zz^101016,zz^131736,0,0,zz^230168,0,zz^53069,zz^145457],
[zz^222890,zz^211013,0,0,zz^31772,zz^136982,zz^50807,zz^131660,0,0,zz^232430,zz^219641,0,zz^37664,zz^243902,0,zz^66272,zz^211139,zz^209309,zz^129494,zz^92600,0,0,zz^9791,0,zz^17951,zz^238223],
[zz^59246,zz^211133,0,0,zz^107663,zz^236687,zz^255833,zz^122054,0,0,zz^78338,zz^188777,0,zz^195512,zz^244619,0,zz^64979,zz^67997,zz^32915,zz^161585,zz^219641,0,0,zz^3674,0,zz^183545,zz^100733],
[zz^183539,zz^149054,0,0,zz^230543,zz^197216,zz^108257,zz^92135,0,0,zz^203228,zz^255833,0,zz^143291,zz^2354,0,zz^183035,zz^29732,zz^189179,zz^150656,zz^50807,0,0,zz^252500,0,zz^164576,zz^128357],
[0,0,zz^88182,zz^100197,0,0,0,0,zz^186597,zz^154467,0,0,zz^43200,0,0,zz^194508,0,0,0,0,0,zz^10800,zz^76788,0,zz^249021,0,0],
[0,0,zz^93582,zz^172800,0,0,0,0,zz^138645,zz^253746,0,0,zz^45009,0,0,zz^209655,0,0,0,0,0,zz^90585,zz^43200,0,zz^222102,0,0],
[zz^53069,zz^50744,0,0,zz^134243,zz^76472,zz^73422,zz^48954,0,0,zz^257829,zz^92391,0,zz^30306,zz^161370,0,zz^250700,zz^64604,zz^42810,zz^133272,zz^188940,0,0,zz^19118,0,zz^67874,zz^23615],
[0,0,zz^103815,zz^101979,0,0,0,0,zz^121176,zz^196776,0,0,zz^209655,0,0,zz^250155,0,0,0,0,0,zz^235224,zz^194508,0,zz^49194,0,0],
[zz^258416,zz^43745,0,0,zz^212276,zz^9353,zz^106062,zz^244152,0,0,zz^168276,zz^145533,0,zz^119475,zz^70404,0,zz^202976,zz^94460,zz^257925,zz^257955,zz^45828,0,0,zz^12686,0,zz^76472,zz^216371],
[zz^188714,zz^79055,0,0,zz^57542,zz^94460,zz^200721,zz^175047,0,0,zz^23400,zz^238986,0,zz^56607,zz^2712,0,zz^101900,zz^35081,zz^64287,zz^23376,zz^119985,0,0,zz^78803,0,zz^64604,zz^178250]];

j1:=GL(27,F)![[zz^260638,zz^49858,zz^251553,zz^145631,zz^10440,zz^141328,zz^147702,zz^234548,zz^192272,zz^168342,zz^49717,zz^81355,zz^246260,zz^232017,zz^174842,zz^112278,zz^6484,zz^260900,zz^101187,zz^186436,zz^150452,zz^204245,zz^166436,zz^253272,zz^137038,zz^48774,zz^12213],
[zz^16936,zz^31738,zz^228920,zz^154257,zz^198850,zz^37883,zz^208189,zz^252480,zz^165846,zz^250817,zz^54164,zz^184907,zz^154038,zz^256682,zz^54466,zz^37514,zz^204840,zz^137406,zz^83850,zz^69003,zz^70360,zz^17135,zz^80407,zz^86041,zz^218909,zz^185391,zz^91101],
[zz^78963,zz^198570,zz^185761,zz^184414,zz^190617,zz^85016,zz^7499,zz^223184,zz^37436,zz^242457,zz^65013,zz^89570,zz^100288,zz^17797,zz^168787,zz^22718,zz^255750,zz^174714,zz^62732,zz^86304,zz^255962,zz^105486,zz^221032,zz^185124,zz^203350,zz^86511,zz^108876],
[zz^240321,zz^39354,zz^146468,zz^106933,zz^198393,zz^29517,zz^250073,zz^128319,zz^3687,zz^24244,zz^218020,zz^84743,zz^203244,zz^68669,zz^99601,zz^245857,zz^206004,zz^6227,zz^73812,zz^252665,zz^123479,zz^153286,zz^48662,zz^85098,zz^223447,zz^162474,zz^126804],
[zz^257171,zz^41026,zz^149082,zz^198611,zz^256123,zz^195096,zz^77522,zz^63277,zz^58238,zz^186969,zz^66522,zz^221458,zz^141458,zz^198868,zz^151763,zz^23866,zz^199432,zz^48852,zz^175082,zz^141639,zz^142605,zz^219783,zz^30551,zz^41760,zz^244802,zz^226659,zz^25936],
[zz^180784,zz^51210,zz^200891,zz^172533,zz^87046,zz^139006,zz^13541,zz^144688,zz^120263,zz^57230,zz^195242,zz^63120,zz^104100,zz^213858,zz^152034,zz^128240,zz^88311,zz^4234,zz^17590,zz^242834,zz^117583,zz^216709,zz^169581,zz^242955,zz^140450,zz^206078,zz^165423],
[zz^19101,zz^65535,zz^130429,zz^130084,zz^187013,zz^138234,zz^174424,zz^5045,zz^246354,zz^171807,zz^173874,zz^79635,zz^231514,zz^166442,zz^68958,zz^76724,zz^35367,zz^237729,zz^235933,zz^241734,zz^91336,zz^75682,zz^215746,zz^372,zz^27481,zz^115953,zz^192759],
[zz^256539,zz^160842,zz^37372,zz^32415,zz^261375,zz^165792,zz^19767,zz^90793,zz^128509,zz^93064,zz^185363,zz^205156,zz^184619,zz^96525,zz^51879,zz^199498,zz^136922,zz^247236,zz^43658,zz^161689,zz^72624,zz^84930,zz^91243,zz^40074,zz^166183,zz^253680,zz^118785],
[zz^24908,zz^118068,zz^96976,zz^26547,zz^174855,zz^125610,zz^231773,zz^76829,zz^165589,zz^196999,zz^213863,zz^224231,zz^194648,zz^85651,zz^251133,zz^107359,zz^157416,zz^245073,zz^136261,zz^12533,zz^33105,zz^61586,zz^88858,zz^7143,zz^14748,zz^78249,zz^37587],
[zz^113190,zz^195009,zz^157443,zz^9359,zz^46281,zz^180714,zz^81789,zz^238804,zz^181909,zz^111976,zz^69985,zz^55796,zz^177175,zz^21576,zz^15683,zz^126150,zz^27219,zz^216348,zz^195062,zz^153464,zz^198482,zz^55258,zz^25072,zz^218235,zz^136751,zz^21254,zz^174750],
[zz^164487,zz^28650,zz^162942,zz^139627,zz^76404,zz^201669,zz^103201,zz^56397,zz^258193,zz^44753,zz^173410,zz^180507,zz^76555,zz^171210,zz^20180,zz^109924,zz^262140,zz^246750,zz^13689,zz^141482,zz^157303,zz^259573,zz^40585,zz^223766,zz^198987,zz^1488,zz^141468],
[zz^261951,zz^165302,zz^152304,zz^97663,zz^141090,zz^171282,zz^242948,zz^209577,zz^238153,zz^9343,zz^89667,zz^88234,zz^204711,zz^105958,zz^141986,zz^23266,zz^95232,zz^260742,zz^18156,zz^51289,zz^201549,zz^219418,zz^242762,zz^63420,zz^180946,zz^41448,zz^61809],
[zz^115134,zz^51501,zz^169393,zz^197529,zz^152346,zz^140910,zz^54505,zz^90436,zz^252469,zz^36617,zz^82703,zz^228687,zz^92269,zz^128702,zz^18453,zz^6061,zz^31701,zz^125616,zz^227477,zz^217793,zz^128054,zz^143237,zz^50811,zz^171690,zz^127000,zz^72915,zz^198164],
[zz^200571,zz^20247,zz^179012,zz^44077,zz^133662,zz^5952,zz^104926,zz^197742,zz^34222,zz^177553,zz^150661,zz^41642,zz^162340,zz^169354,zz^225588,zz^9519,zz^114600,zz^41586,zz^80720,zz^160554,zz^54756,zz^127482,zz^251863,zz^43473,zz^246343,zz^108635,zz^262131],
[zz^202515,zz^138882,zz^110113,zz^214190,zz^239727,zz^228291,zz^28353,zz^34195,zz^129660,zz^11563,zz^79068,zz^122470,zz^102829,zz^217166,zz^101029,zz^140446,zz^119082,zz^212997,zz^207516,zz^123957,zz^174632,zz^149488,zz^77577,zz^259071,zz^251893,zz^160296,zz^23402],
[zz^77106,zz^203412,zz^144886,zz^111013,zz^251166,zz^114288,zz^83032,zz^200528,zz^230795,zz^235968,zz^5394,zz^59701,zz^198947,zz^38366,zz^179837,zz^27994,zz^174759,zz^159369,zz^180692,zz^13949,zz^85983,zz^6268,zz^240901,zz^136385,zz^162609,zz^176250,zz^54087],
[zz^25338,zz^151532,zz^216839,zz^91866,zz^67744,zz^217278,zz^19297,zz^215342,zz^92742,zz^150056,zz^46327,zz^13869,zz^59485,zz^216656,zz^223491,zz^89207,zz^126952,zz^102261,zz^217864,zz^240299,zz^73257,zz^129251,zz^68540,zz^8971,zz^139098,zz^82021,zz^32931],
[zz^2610,zz^1621,zz^116597,zz^48068,zz^63318,zz^143536,zz^77965,zz^174782,zz^165331,zz^128424,zz^123540,zz^58637,zz^233015,zz^46609,zz^221904,zz^173157,zz^68589,zz^196231,zz^37613,zz^216946,zz^167997,zz^41609,zz^61565,zz^143265,zz^159141,zz^35332,zz^65225],
[zz^65559,zz^126735,zz^46252,zz^149173,zz^23631,zz^116898,zz^174242,zz^227737,zz^70331,zz^37498,zz^113412,zz^233685,zz^48165,zz^54129,zz^136780,zz^221143,zz^31242,zz^93608,zz^141973,zz^82235,zz^43635,zz^178309,zz^73666,zz^172479,zz^256497,zz^249855,zz^214185],
[zz^166344,zz^23808,zz^185926,zz^125074,zz^15855,zz^172397,zz^219024,zz^166568,zz^176308,zz^38076,zz^157561,zz^117930,zz^221023,zz^78358,zz^4539,zz^198943,zz^80988,zz^262095,zz^115923,zz^153130,zz^60737,zz^191762,zz^247785,zz^10362,zz^136888,zz^173892,zz^196257],
[zz^112289,zz^205449,zz^149992,zz^192660,zz^93,zz^212991,zz^174540,zz^148311,zz^72406,zz^98143,zz^172682,zz^66797,zz^32521,zz^191505,zz^124519,zz^239559,zz^244797,zz^70311,zz^22834,zz^216516,zz^43606,zz^185008,zz^188950,zz^94524,zz^19181,zz^165630,zz^124968],
[zz^174570,zz^77921,zz^183399,zz^139009,zz^53709,zz^83901,zz^237419,zz^96137,zz^213370,zz^90872,zz^29996,zz^83073,zz^97699,zz^260052,zz^106307,zz^26971,zz^7851,zz^173361,zz^150862,zz^71188,zz^250928,zz^218758,zz^159801,zz^238182,zz^149744,zz^216210,zz^236571],
[zz^169158,zz^73461,zz^101345,zz^128653,zz^173994,zz^78411,zz^217283,zz^70149,zz^31750,zz^107884,zz^163247,zz^22609,zz^114918,zz^119984,zz^122405,zz^74690,zz^49541,zz^159855,zz^163085,zz^253779,zz^79162,zz^209310,zz^88603,zz^214836,zz^67051,zz^166299,zz^31404],
[zz^195408,zz^256098,zz^223590,zz^41546,zz^242255,zz^120207,zz^46134,zz^99403,zz^8015,zz^95464,zz^47945,zz^42270,zz^122204,zz^3945,zz^253108,zz^192779,zz^164104,zz^103744,zz^82766,zz^9043,zz^176042,zz^72042,zz^92703,zz^238063,zz^232952,zz^167040,zz^11299],
[zz^193863,zz^240297,zz^1567,zz^254306,zz^99632,zz^50853,zz^132420,zz^110495,zz^106188,zz^167293,zz^140663,zz^50132,zz^93289,zz^69023,zz^45173,zz^58992,zz^210129,zz^150348,zz^218103,zz^80461,zz^20758,zz^125761,zz^246344,zz^175134,zz^138070,zz^28572,zz^105378],
[zz^152833,zz^218685,zz^119713,zz^226673,zz^257346,zz^143874,zz^179882,zz^169080,zz^166184,zz^246830,zz^184536,zz^36172,zz^108669,zz^191780,zz^135469,zz^145379,zz^237963,zz^45196,zz^226003,zz^15780,zz^68921,zz^107931,zz^26025,zz^182591,zz^32060,zz^165823,zz^132130],
[zz^146901,zz^82683,zz^75938,zz^237940,zz^101352,zz^65941,zz^30885,zz^55476,zz^105321,zz^94685,zz^77188,zz^174767,zz^12017,zz^185308,zz^74939,zz^32106,zz^81842,zz^131724,zz^107535,zz^80195,zz^85027,zz^80927,zz^254861,zz^8833,zz^108825,zz^35884,zz^245665]];

j2:=GL(27,F)![[zz^15869,zz^20513,zz^59034,zz^99285,zz^75766,zz^24929,zz^80421,zz^14325,zz^170032,zz^19677,zz^69441,zz^163839,zz^228576,zz^194439,zz^141195,zz^156822,zz^128049,zz^25605,zz^11904,zz^233796,zz^82651,zz^251220,zz^101706,zz^172413,zz^167802,zz^131882,zz^240414],
[zz^99425,zz^259133,zz^218499,zz^226380,zz^33872,zz^5220,zz^261759,zz^38202,zz^157926,zz^230268,zz^250935,zz^224578,zz^154212,zz^142887,zz^66831,zz^76173,zz^252199,zz^43523,zz^138999,zz^131118,zz^70545,zz^49816,zz^125583,zz^50676,zz^86997,zz^31659,zz^128673],
[zz^82923,zz^29119,zz^213866,zz^18718,zz^46371,zz^96136,zz^195326,zz^260168,zz^106685,zz^132915,zz^64830,zz^123177,zz^222026,zz^166237,zz^17111,zz^257306,zz^135079,zz^191203,zz^88154,zz^36203,zz^250148,zz^53094,zz^246469,zz^183732,zz^15875,zz^213737,zz^83092],
[zz^114460,zz^74541,zz^48488,zz^223952,zz^239491,zz^256848,zz^18686,zz^81471,zz^222771,zz^73234,zz^186128,zz^196286,zz^209793,zz^23126,zz^89506,zz^215768,zz^111795,zz^231517,zz^92963,zz^74996,zz^76152,zz^131855,zz^72443,zz^37969,zz^181744,zz^189370,zz^190928],
[zz^102420,zz^99716,zz^78708,zz^127875,zz^63476,zz^3242,zz^68461,zz^131070,zz^134997,zz^103002,zz^59541,zz^148755,zz^144681,zz^15621,zz^57300,zz^146922,zz^82052,zz^175227,zz^40494,zz^253470,zz^47616,zz^236136,zz^218451,zz^40921,zz^155842,zz^165366,zz^250053],
[zz^8468,zz^259657,zz^12454,zz^170553,zz^12669,zz^130319,zz^259341,zz^213315,zz^87285,zz^251232,zz^232329,zz^140622,zz^56595,zz^163851,zz^231357,zz^57567,zz^97704,zz^90392,zz^83172,zz^187216,zz^262047,zz^228003,zz^38553,zz^204522,zz^84579,zz^1305,zz^207488],
[zz^126240,zz^162710,zz^169486,zz^111592,zz^107671,zz^117274,zz^176468,zz^159270,zz^179140,zz^195231,zz^148169,zz^133594,zz^119402,zz^244940,zz^98871,zz^45218,zz^180773,zz^72344,zz^83284,zz^205227,zz^235860,zz^186319,zz^100264,zz^27738,zz^166146,zz^87391,zz^84540],
[zz^27082,zz^33261,zz^238003,zz^163578,zz^154235,zz^155930,zz^223753,zz^86705,zz^14998,zz^109010,zz^39534,zz^86937,zz^166064,zz^56706,zz^206402,zz^172423,zz^155044,zz^97621,zz^209852,zz^86341,zz^175905,zz^201403,zz^2697,zz^38594,zz^212695,zz^61770,zz^92268],
[zz^139639,zz^240963,zz^30793,zz^52743,zz^195697,zz^233194,zz^42465,zz^260858,zz^109379,zz^76643,zz^74744,zz^37841,zz^27629,zz^220226,zz^63741,zz^202690,zz^36021,zz^239426,zz^95881,zz^92504,zz^109709,zz^193952,zz^3134,zz^171535,zz^104655,zz^151876,zz^185037],
[zz^208200,zz^230377,zz^144345,zz^92207,zz^45933,zz^203887,zz^147279,zz^200885,zz^200576,zz^184538,zz^107095,zz^65042,zz^135751,zz^205658,zz^153110,zz^229836,zz^20773,zz^217338,zz^62537,zz^96330,zz^179903,zz^127153,zz^186578,zz^118970,zz^195398,zz^24034,zz^244408],
[zz^27233,zz^206953,zz^256638,zz^215465,zz^242817,zz^87421,zz^157011,zz^10090,zz^184225,zz^180872,zz^181586,zz^34479,zz^138913,zz^68390,zz^112794,zz^140298,zz^126554,zz^76017,zz^133341,zz^193331,zz^70993,zz^153658,zz^220990,zz^168541,zz^192274,zz^110952,zz^198806],
[zz^235166,zz^38761,zz^246958,zz^134821,zz^140720,zz^73851,zz^140955,zz^182672,zz^249781,zz^256108,zz^145248,zz^87212,zz^171966,zz^87121,zz^52463,zz^158324,zz^23067,zz^137842,zz^109512,zz^87270,zz^121474,zz^66210,zz^41516,zz^146514,zz^239713,zz^170054,zz^89941],
[zz^256480,zz^224556,zz^229571,zz^252300,zz^75028,zz^84171,zz^46532,zz^153448,zz^45436,zz^12122,zz^136853,zz^216975,zz^55988,zz^18749,zz^219848,zz^149380,zz^47732,zz^28615,zz^19038,zz^180143,zz^135743,zz^214718,zz^117984,zz^178414,zz^53942,zz^64212,zz^123415],
[zz^41925,zz^87541,zz^199202,zz^31366,zz^108932,zz^181665,zz^21829,zz^137916,zz^75431,zz^36906,zz^103758,zz^249038,zz^97531,zz^202058,zz^40360,zz^244810,zz^41383,zz^8795,zz^189033,zz^11417,zz^9078,zz^240123,zz^90346,zz^184839,zz^212614,zz^149878,zz^244073],
[zz^128341,zz^99434,zz^173897,zz^139970,zz^108328,zz^247080,zz^179334,zz^85605,zz^130026,zz^165406,zz^108583,zz^83221,zz^10788,zz^158136,zz^84677,zz^64351,zz^133044,zz^106929,zz^39179,zz^226824,zz^52979,zz^165583,zz^19183,zz^92654,zz^59992,zz^154376,zz^95890],
[zz^77019,zz^70729,zz^97324,zz^50144,zz^160814,zz^123130,zz^223381,zz^169349,zz^179921,zz^101622,zz^182486,zz^115757,zz^219659,zz^155154,zz^81170,zz^177206,zz^61102,zz^52050,zz^241583,zz^147332,zz^233427,zz^177716,zz^230545,zz^137080,zz^57459,zz^247579,zz^185406],
[zz^174092,zz^20880,zz^134643,zz^92562,zz^135557,zz^126636,zz^20037,zz^111883,zz^119091,zz^42549,zz^260607,zz^186,zz^240189,zz^217311,zz^152808,zz^85845,zz^250103,zz^252549,zz^5181,zz^47262,zz^31710,zz^87567,zz^199264,zz^135488,zz^107418,zz^202704,zz^222367],
[zz^150013,zz^97548,zz^62805,zz^42508,zz^108639,zz^70664,zz^82896,zz^231906,zz^173022,zz^145830,zz^245217,zz^69117,zz^90357,zz^58449,zz^2976,zz^70455,zz^191175,zz^69503,zz^217270,zz^237567,zz^85641,zz^156498,zz^57144,zz^164042,zz^170277,zz^71768,zz^71937],
[zz^165573,zz^201891,zz^137338,zz^43152,zz^251221,zz^93218,zz^211916,zz^70741,zz^35594,zz^257404,zz^193050,zz^120867,zz^76732,zz^172189,zz^80277,zz^239968,zz^135593,zz^121417,zz^76565,zz^108258,zz^156716,zz^171302,zz^138046,zz^171169,zz^257961,zz^108473,zz^7890],
[zz^35180,zz^202374,zz^147624,zz^127981,zz^167700,zz^75226,zz^36312,zz^209723,zz^125464,zz^192811,zz^87316,zz^45668,zz^99241,zz^152889,zz^27378,zz^64027,zz^88021,zz^189863,zz^161440,zz^21803,zz^231846,zz^10379,zz^174063,zz^173585,zz^39581,zz^215070,zz^165532],
[zz^223525,zz^110729,zz^243187,zz^44785,zz^138006,zz^171749,zz^102578,zz^221325,zz^172608,zz^173443,zz^61235,zz^170889,zz^27898,zz^247914,zz^20821,zz^245415,zz^21135,zz^31560,zz^58965,zz^164470,zz^44117,zz^25066,zz^160922,zz^218455,zz^142376,zz^160390,zz^18086],
[zz^240526,zz^122401,zz^7374,zz^101675,zz^69549,zz^68519,zz^214163,zz^230565,zz^74872,zz^242795,zz^257018,zz^144812,zz^199447,zz^259320,zz^254243,zz^63500,zz^116476,zz^70225,zz^68444,zz^140662,zz^90473,zz^69035,zz^212376,zz^185484,zz^164597,zz^210642,zz^16030],
[zz^18757,zz^11933,zz^184751,zz^11359,zz^175675,zz^56139,zz^99749,zz^54962,zz^144557,zz^254000,zz^70223,zz^38362,zz^63075,zz^241643,zz^135831,zz^134102,zz^227461,zz^64120,zz^230543,zz^250851,zz^11633,zz^29496,zz^13997,zz^16053,zz^37345,zz^217650,zz^203761],
[zz^176622,zz^12968,zz^149865,zz^54438,zz^147537,zz^137178,zz^190464,zz^70734,zz^249357,zz^63402,zz^11701,zz^227451,zz^87375,zz^238164,zz^262137,zz^99082,zz^136721,zz^213783,zz^229200,zz^62484,zz^161976,zz^52689,zz^158115,zz^253904,zz^15702,zz^163684,zz^66065],
[zz^171275,zz^146347,zz^44429,zz^110516,zz^34270,zz^83218,zz^176693,zz^151364,zz^210972,zz^24331,zz^169860,zz^107873,zz^12536,zz^36833,zz^257003,zz^156477,zz^177423,zz^215862,zz^254964,zz^94475,zz^121381,zz^123172,zz^251522,zz^258502,zz^175373,zz^161854,zz^144084],
[zz^68703,zz^24426,zz^253608,zz^87357,zz^182202,zz^130450,zz^123618,zz^123375,zz^217752,zz^134185,zz^237570,zz^249936,zz^108174,zz^46804,zz^20793,zz^62808,zz^51872,zz^2117,zz^262119,zz^166227,zz^130371,zz^75174,zz^210756,zz^65862,zz^210999,zz^229187,zz^22598],
[zz^223767,zz^244401,zz^170196,zz^174327,zz^172082,zz^24387,zz^126840,zz^744,zz^108105,zz^81237,zz^80148,zz^189048,zz^10627,zz^255999,zz^185389,zz^167529,zz^83520,zz^103039,zz^86946,zz^82815,zz^20724,zz^14286,zz^88125,zz^17942,zz^214221,zz^17666,zz^213983]];

J1:=sub<GL(27,F)|l1,l2,j1>;
J2:=sub<GL(27,F)|l1,l2,j2>;
// Here is the hom space for L, even as a subgroup of NGT. This makes computation incredibly easy

R<a,m1,m2,m3,m4>:=PolynomialRing(F,5);
mats:=MatrixRing(R,27);
scal:=mats!DiagonalMatrix([a,a,m1,m1,a,a,m3,m3,m1,m1,m3,m3,m1,m3,m3,m1,a,a,m3,m3,m3,m1,m1,a,m1,a,a]);
a1:=[3,22,23,13,4,9,25,16,10];
a2:=[20,12,8,15,19,21,7,11,14];
for i in [1..9] do scal[a1[i],a2[i]]:=m2; scal[a2[i],a1[i]]:=m4; end for;

seqs:=[[i,j,k]:i,j,k in [1..NumberOfRows(l1)]|i le j and j le k];

ents:=[300,322,340,354,359,609,637,661,681,697,901,917,947,1004,1020,1164,1182,1246,1283,1311,1399,1419,1521,1558,1577,1710,1744,1787,
1821,1922,1958,1989,2029,2184,2227,2260,2357,2388,2427,2586,2609,2710,2749,2859,2991];

f27:=[0:i in [1..#seqs]];
for i in [1..#ents] do f27[ents[i]]:=1; end for;

function Matricise(v)
return Matrix(1,NumberOfColumns(v),[R!v[i]:i in [1..NumberOfColumns(v)]]);
end function;

V:=GModule(L);
g1:=mats!h1;
g2:=mats!h2;

intseqs1:=[seqs[i]:i in ents];
intseqs:=&join{{[i[1],i[2],i[3]],[i[1],i[3],i[2]],[i[2],i[1],i[3]],[i[2],i[3],i[1]],[i[3],i[1],i[2]],[i[3],i[2],i[1]]}:i in intseqs1};

function ProdRel(seq,g)

u:=V.seq[1]; v:=V.seq[2]; w:=V.seq[3];
u1:=Matricise(u)*scal; v1:=Matricise(v)*scal; w1:=Matricise(w)*scal;
u2:=Matricise(u)*g*scal; v2:=Matricise(v)*g*scal; w2:=Matricise(w)*g*scal;
e1:=&+[u1[1,i[1]]*v1[1,i[2]]*w1[1,i[3]]*f27[Position(seqs,Sort([i[1],i[2],i[3]]))]:i in intseqs];
e2:=&+[u2[1,i[1]]*v2[1,i[2]]*w2[1,i[3]]*f27[Position(seqs,Sort([i[1],i[2],i[3]]))]:i in intseqs];

return e1-e2;
end function;

function CheckRel(seq,gg)
u:=V.seq[1]; v:=V.seq[2]; w:=V.seq[3];
u2:=u^gg; v2:=v^gg; w2:=w^gg;
e1:=&+[u[i[1]]*v[i[2]]*w[i[3]]*f27[Position(seqs,Sort([i[1],i[2],i[3]]))]:i in intseqs];
e2:=&+[u2[i[1]]*v2[i[2]]*w2[i[3]]*f27[Position(seqs,Sort([i[1],i[2],i[3]]))]:i in intseqs];
return e1-e2;
end function;

coeffs0:=[a,m1,m2,m3,m4];

function ChangeCoefficient(coeffs,coeff,target)
coeffs2:=coeffs;
nn:=Position(coeffs0,coeff);
coeffs2[nn]:=target;
return [Evaluate(i,coeffs2):i in coeffs];
end function;

function ChangeCoefficients(coeffs,coeff,target)
coeffs2:=coeffs;
for i in [1..#coeff] do
  nn:=Position(coeffs0,coeff[i]);
  coeffs2[nn]:=target[i];
end for;
return [Evaluate(i,coeffs2):i in coeffs];
end function;

// We now check that there is a unique H over L for each module action.

function CheckDetermination()

ProdRel([1,5,20],g1) eq zz^213614*a*(a+zz^246286*m4)^2;
ProdRel([1,5,20],g2) eq zz^243334*a*(a+zz^41804*m4)^2;
c11:=[a,m1,m2,m3,a/zz^246286];
c21:=[a,m1,m2,m3,a/zz^41804];

Evaluate(ProdRel([1,5,21],g1),c11) eq zz^209015*a^2*(a+zz^158905*m3);
Evaluate(ProdRel([1,5,21],g2),c21) eq zz^151354*a^2*(a+zz^129185*m3);
c12:=ChangeCoefficient(c11,m3,a/zz^158905);
c22:=ChangeCoefficient(c21,m3,a/zz^129185);

Evaluate(ProdRel([1,3,24],g1),c12) eq zz^4599*a^2*(m1+zz^87381*m2);
Evaluate(ProdRel([1,3,23],g2),c22) eq zz^82782*a*(m1+zz^174762*m2)^2;
c13:=ChangeCoefficient(c12,m2,m1/zz^87381);
c23:=ChangeCoefficient(c22,m2,m1/zz^174762);

// Now remove the centralizer of H in GL(27,k):

c14:=ChangeCoefficients(c13,[a,m1],[1,1]);
c24:=ChangeCoefficients(c23,[a,m1],[1,1]);

h11:=h1^(GL(27,F)!Evaluate(scal,c14));
h22:=h2^(GL(27,F)!Evaluate(scal,c24));

H1:=sub<GL(27,F)|l1,l2,h11>;
H2:=sub<GL(27,F)|l1,l2,h22>;

// As the two 18-dimensional factors here are not Aut(H)-conjugate, one obtains two classes of H
// containing the given L. (Note that the two dual 9s are Aut(H)-conjugate.)

h11 in J1; h22 in J2;

// Now check that J1 and J2 lie in E6.

{CheckRel(i,j1):i in seqs} eq {0};
{CheckRel(i,j2):i in seqs} eq {0};

// This proves that both of the copies of PSL(2,19) we constructed are contained in PSL(2,19)<J3.

return "";
end function;

// We now give code that can prove that the trilinear form we gave is the correct one. Choose two random elements that
// generate the subgroup 19^6.W(E6), and then build the trilinear form relations. We also check that l1 and l2 lie in
// NGT.

function CheckE6Form()

q:=2^18;
GG:=GroupOfLieType("E6",q);
W:=VectorSpace(F,6);
rho:=StandardRepresentation(GG);
Over:=GL(27,q);
g1:=elt<GG|W![z,1,1,1,1,1]>;
g2:=elt<GG|W![1,z,1,1,1,1]>;
g3:=elt<GG|W![1,1,z,1,1,1]>;
g4:=elt<GG|W![1,1,1,z,1,1]>;
g5:=elt<GG|W![1,1,1,1,z,1]>;
g6:=elt<GG|W![1,1,1,1,1,z]>;
Refs:=Reflections(GG);
Mats:=[rho(i):i in [g1,g2,g3,g4,g5,g6]];
MatsE:=[rho(i):i in Refs];
NGT:=sub<Over|Mats cat MatsE>;
T:=sub<NGT|Mats>;
E:=sub<NGT|MatsE>;

l1 in NGT and l2 in NGT;

repeat y1:=Random(NGT); y2:=Random(NGT); until NGT eq sub<NGT|y1,y2>;

mat:=[];
for h in [y1,y2] do
  for nn in [1..#seqs] do aa:=seqs[nn,1]; bb:=seqs[nn,2]; cc:=seqs[nn,3];
    val:=[F!0:i in [1..#seqs]];
    for i in [1..NumberOfRows(l1)] do if(h[aa,i] ne 0) then for j in [1..NumberOfRows(l1)] do if(h[bb,j] ne 0) then for k in [1..NumberOfRows(l1)] do if(h[cc,k] ne 0) then
      val[Position(seqs,Sort([i,j,k]))]+:=h[aa,i]*h[bb,j]*h[cc,k];
    end if; end for; end if; end for; end if; end for;
    val[nn]-:=1; Append(~mat,val); delete val;
  end for;
end for;

ftest:=Nullspace(Transpose(Matrix(F,mat))).1;
return &and[ftest[i] eq f27[i]:i in [1..#seqs]];
end function;

"The function CheckE6Form() checks that the trilinear form f27 from E6 is correct";
"The function CheckDetermination() checks that the determination of the possible conjugates into E6 is correct.";
